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15   THE LEBESGUE-NIKODYM THEOREM       193

are locally /l-integrable real-valued functions. Hence the assertions of (i) are
consequences of those of (ii).

To prove (13.15.3.3), we reduce to the case where n2 = Q; for if
supO*! - yU2 , 0) exists and is equal to (gt - g2)+ - ^ it follows immediately
(13.3) that we shall have

A*2 > 0) = (g2 + (gl -~ #2)+)  A = sup(^, g2) - L

This reduces us to proving (13.15.3.4). For this, we shall begin by proving the
last assertion of (ii), or equivalently that the relation^ * A ^ 0 implies g(x) ^ 0
almost everywhere with respect to L Let N be the set of points x e X such that
g(x) < 0. If we put v = $~A, it follows from (13.14.1.5) that v(X - N) =0;
so if we can show that v(N) s= 0, it will follow that v = 0, and this will estab-
lish our assertion (13.14.4). Clearly it is enough to prove that v(N n K) =0
for each compact subset K of X. If U is any relatively compact open set

containing N n K, then we have j   g~ dl <S f g+ dL For by hypothesis

/                                                                           J u                     J u

J(#+ -g~)fdk ^ 0 for every function/^ 0 belonging to 2fR(X), and it is
therefore sufficient to remark that | g~ dX =sup f g""fdk, ( g+ dl =

/                                                                                                 J U                                     J                        J U

supl g+fdA, the supremum being taken over all functions /e JfR(X) such
that 0 g/^ ^,j (13.5.1). Since #+(x) = 0 for x e N n K, we have

I       ^+^ = inf f ^+^-0;

JNnK                U   Ju

consequently, f       g   dk = inf f' g   dk= 0, which shows that v(N n K) = 0

JNnK                  y   J U

(13.14.3).

Now let p be a measure such that p ^ 0 and p ^ g  A. Putting A + p = o-,
we have (13.15.1) p = w  a and A = P  <r, where w and y are locally cr-integrable
and u g 0 and t? ^ 0 almost everywhere with respect to cr. It follows that
u^gv almost everywhere with respect to <j, hence that u^(gv)* =g*v
almost everywhere with respect to <r; but this implies that

p = u - cr ^ (^+f) * a = #* -A

by virtue of (13.14.5).

Finally, we have to prove (iii). If the sequence (gn - X) is bounded above in
MR(X), then for each function/^ 0 belonging to ^R(X) we have

sup \fgndX< +00;

n    Jlies that v*(/z) ^ /z(/z) ^ e, and the proof is complete.